Smooth Fano Polytopes Arising from Finite Partially Ordered Sets

Gorenstein Fano polytopes arising from finite partially ordered sets will be introduced. Then we study the problem which partially ordered sets yield smooth Fano polytopes. Introduction An integral (or lattice) polytope is a convex polytope all of whose vertices have integer coordinates. Let P ⊂ R be an integral convex polytope of dimension d. • We say that P is a Fano polytope if the origin of R is a unique integer point belonging to the interior of P. • A Fano polytope P is called terminal if each integer point belonging to the boundary of P is a vertex of P. • A Fano polytope P is called canonical if P is not terminal, i.e., there is an integer point belonging to the boundary of P which is not a vertex of P. • A Fano polytope is called Gorenstein if its dual polytope is integral. (Recall that the dual polytope P of a Fano polytope P is the convex polytope which consists of those x ∈ R such that 〈x, y〉 ≤ 1 for all y ∈ P, where 〈x, y〉 is the usual inner product of R.) • A Q-factorial Fano polytope is a simplicial Fano polytope, i.e., a Fano polytope each of whose faces is a simplex. • A smooth Fano polytope is a Fano polytope such that the vertices of each facet are a Z-basis of Z. Thus in particular a smooth Fano polytope is Q-factorial Fano, Gorenstein and terminal. Øbro [4] succeeded in finding an algorithm which yields the classification list of the smooth Fano polytopes for given d. It is proved in Casagrande [1] that the number of vertices of a Gorenstein Q-factorial Fano polytope is at most 3d if d is even, and at most 3d−1 if d is odd. B. Nill and M. Øbro [2] classified the Gorenstein Q-factorial Fano polytopes of dimension d with 3d − 1 vertices. The study on the classification of terminal or canonical Fano polytopes was done by Kasprzyk [3]. 2000 Mathematics Subject Classification: Primary 14J45, 52B20; Secondary 06A11.